Interpretability logic

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities.

Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella.

Examples

edit

Logic ILM

edit

The language of ILM extends that of classical propositional logic by adding the unary modal operator   and the binary modal operator   (as always,   is defined as  ). The arithmetical interpretation of   is “  is provable in Peano arithmetic (PA)”, and   is understood as “  is interpretable in  ”.

Axiom schemata:

  1. All classical tautologies
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  
  8.  
  9.  

Rules of inference:

  1. “From   and   conclude  
  2. “From   conclude  ”.

The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov.

Logic TOL

edit

The language of TOL extends that of classical propositional logic by adding the modal operator   which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of   is “  is a tolerant sequence of theories”.

Axioms (with   standing for any formulas,   for any sequences of formulas, and   identified with ⊤):

  1. All classical tautologies
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  

Rules of inference:

  1. “From   and   conclude  
  2. “From   conclude  ”.

The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze.

References

edit